Again, multiplying by Aw1, Bw,, Ca,, adding, and integrating, we have

2

2

2

A2∞,2 + B2∞,2 + C2w ̧2 = k2,

where k2 is the constant of integration.

From these two equations we obtain

C

dw. {k3 — Bh + (B — C') Cw ̧3}* { − k2+ Ah + (C − A) Cœ ̧”}*

dt

3

whence t+?

=

3

√(AB)

dw3

where 7 is the constant of integration.

This integral cannot in the general case be found; we may however approximate: thus t is known in terms of w,, and consequently w, in terms of t; and then from above, w1, w, are also known.

4. With respect to the constants introduced by the integration, we may remark that h represents the vis viva (Routh's Rigid Dynamics, Art. 194), and k the area conserved on the invariable plane. To prove the latter point, the areas conserved on the principal planes being Aw1, Bw2, Cw ̧ (Routh's R. D., Art 179), and the direction cosines of the invariable plane with reference to the principal axes

(Routh's R. D., Art. 125), the area conserved on the invari

5. When w1, w2, w, are known at any time, the resultant angular velocity of the planet is known, and also the position of the instantaneous axis of rotation with reference to the principal axes. It remains to shew how the position of these axes in space may be determined.

Suppose a sphere described with its centre at the centre of gravity of the planet and its radius of any magnitude: take as a plane of reference any fixed plane passing through the centre of gravity, and let it cut the sphere in the great circle ON; also let the principal plane of xy cut the sphere in the great circle NAB, N being the node of this plane upon the fixed plane, and A, B, C the points where the sphere is cut by the principal axes of x, y, z. Take P the pole of ON, and join PA, PB, CA, CB by arcs of great circles.

P

N

A

B

Let the angle ONA=0, ON=†, NA=4: then if the

angles 0, 4, ↓ be known, the position of the planet will be determined with reference to the fixed plane.

Now we may consider the planet to be moving with anguabout the principal axes; or with

lar velocities, w1, w,, w,

the fixed plane, the second about the line of nodes, the third about the principal axis of z. We shall adopt the usual convention with respect to signs, and consider positive those angular velocities which tend to turn the planet round the axes of x, y, z from y to z, z to x, x to y, respectively.

Thus, resolving about the principal axes, we have

By substituting the values of w,, w,, w, obtained as above (Art. 3), and then integrating these equations, 0, 4, and would be determined, and thus the position of the principal axes at any time would be known. The integration, however, cannot in general be effected; so that we are obliged to have recourse to a special hypothesis with regard to the position of the fixed plane of reference. If we take for this purpose the invariable plane, the process becomes much simplified.

6. Let then 1, 1, 1 denote relatively to the invariable plane the same angles which relatively to the original plane of reference have been denoted by 0, p, . Then, the direction cosines of the invariable plane with reference to the principal axes being

= cos PA

respectively, we have (see figure of preceding Article)

Αω,

k

* There is much disagreement between writers as to the measurement of the angles employed in these kinematical equations; the above, however, agrees with La Place, Poisson, and Pontécoulant.

These equations give 0, and 4,: to obtain 1, substitute in the first of the equations (A) of Art. 5; thus

combining this with the result of Art. 3, and integrating, we have

Y1+g=−kС√(AB) ×
(h-Cw)dw3

(k3 — C3w ̧3){h3 — Bh+(B−C) Cw ̧2}1 {− k2+Ah+(C−A) Cw ̧2} › where g is the constant of integration.

Since, is known in terms of t from Art. 3, these equations give 01, 1, 1; so that the position of the principal axes is known at any time with reference to the invariable plane. Since, however, when the disturbing forces are taken into account, this plane ceases to be absolutely invariable, it will be convenient to be able to refer the motion to some other plane which does remain fixed, and which may be taken as a plane of reference: this we can now do by Spherical Trigonometry.